What does this mean: "I'll be with you in a minute"? Does it mean "I'll call you after one minute" or "I'll contact you after one minute" or something else?
Well usually this is used when speaking face-to-face, and is taken to mean
Not exactly one minute, but a short enough time. Similar usage to "Just a second" or "Gimme a sec"
As @Martha says - On the phone, this would likely mean you should hold. The correct statement if he intends to ring you back would be
As JoseK said, this is most often used in person, and it means something like "hold on, let me finish this, and then I'll be able to help you/talk to you."
If someone said this to me on the phone, I would definitely interpret it as "please hold", not "I'll call you back", and most certainly not "Please call back in a minute".